We consider the following generalization of the classic Binary Search Problem: a searcher is required to find a hidden target vertex $x$ in a graph $G$, by iteratively performing queries about vertices. A query to $v$ incurs a cost $c(v, x)$ and responds whether $v=x$ and if not, returns the connected component in $G-v$ containing $x$. The goal is to design a search strategy that minimizes the average-case search cost. Firstly, we consider the case when the cost of querying a vertex is independent of the target. We develop a $\br{4+ε}$-approximation FPTAS for trees running in $O(n^4/ε^2)$ time and an $O({\sqrt{\log n}})$-approximation for general graphs. Additionally, we give an FPTAS parametrized by the number of non-leaf vertices of the graph. On the hardness side we prove that the problem is NP-hard even when the input is a tree with bounded degree or bounded diameter. Secondly, we consider trees and assume $c(v, x)$ to be a monotone non-decreasing function with respect to $x$, i.e.\ if $u \in P_{v, x}$ then $c(u, x) \leq c(v, x)$. We give a $2$-approximation algorithm which can also be easily altered to work for the worst-case variant. This is the first constant factor approximation algorithm for both criterions. Previously known results only regard the worst-case search cost and include a parametrized PTAS as well as a $4$-approximation for paths. At last, we show that when the cost function is an arbitrary function of the queried vertex and the target, then the problem does not admit any constant factor approximation under the UGC, even when the input tree is a star.

翻译：我们考虑经典二分搜索问题的以下推广：搜索者需要在一张图 $G$ 中寻找一个隐藏的目标顶点 $x$，通过反复对顶点进行查询。对 $v$ 的查询会产生代价 $c(v, x)$，并返回 $v=x$ 是否成立，若不成立，则返回 $G-v$ 中包含 $x$ 的连通分量。目标是设计一种搜索策略，以最小化平均情况下的搜索代价。首先，我们考虑查询顶点代价与目标无关的情形。我们针对树提出了一种运行时间为 $O(n^4/ε^2)$ 的 $\br{4+ε}$-近似FPTAS，以及针对一般图的一种 $O({\sqrt{\log n}})$-近似算法。此外，我们给出了一种以图中非叶节点数量为参数的FPTAS。在困难性方面，我们证明了即使输入是度有界或直径有界的树，该问题也是NP-难的。其次，我们考虑树，并假设 $c(v, x)$ 是关于 $x$ 的单调非递减函数，即如果 $u \in P_{v, x}$，则 $c(u, x) \leq c(v, x)$。我们给出了一种$2$-近似算法，该算法也可方便地修改以适用于最坏情况变体。这是针对这两种准则的首个常数因子近似算法。先前已知的结果仅关注最坏情况搜索代价，包括一种参数化PTAS以及针对路径的一种$4$-近似算法。最后，我们证明当代价函数是查询顶点和目标的任意函数时，在UGC假设下该问题不允许任何常数因子近似，即使输入树是一颗星图。